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Question
Tangent segments drawn from an external point to a circle are congruent, prove this theorem. Complete the following activity.
Given: `square`
To Prove: `square`
Proof: Draw radius AP and radius AQ and complete the following proof of the theorem.
In ∆PAD and ∆QAD,
seg PA ≅ `square` .....[Radii of the same circle]
seg AD ≅ seg AD ......[`square`]
∠APD ≅ ∠AQD = 90° .....[Tangent theorem]
∴ ∆PAD ≅ ∆QAD ....[`square`]
∴ seg DP ≅ seg DQ .....[`square`]
Solution
Given: A is the centre of the circle. Tangents through external point D Touch the circle at the points P and Q.
To Prove: seg DP ≅ seg DQ
Proof:
In ∆PAD and ∆QAD,
seg PA ≅ seg QA .....[Radii of the same circle]
seg AD ≅ seg AD ......[Common side]
∠APD = ∠AQD = 90° .....[Tangent theorem]
∴ ∆PAD ≅ ∆QAD .....[By Hypotenuse side test]
∴ seg DP ≅ seg DQ .....[Corresponding sides of congruent triangles]
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